Stable reduction of $X_0(p^4)$

TL;DR

This paper extends the computation of stable reductions from $X_0(p^3)$ to $X_0(p^4)$, revealing new irreducible components related to Deligne-Lusztig curves and detailing intersection multiplicities.

Contribution

It provides the first explicit stable reduction of $X_0(p^4)$, identifying new components and intersection data based on prior methods.

Findings

Irreducible components defined by $a^p - a = t^{p+1}$

Components are Deligne-Lusztig curves for ${ m SL}_2(F_p)$

Computed intersection multiplicities in the stable reduction

Abstract

R. Coleman and K. McMurdy compute the stable reduction of $X_0(p^3).$ On the basis of their ideas, we compute the stable reduction of $X_0(p^4).$ As a result, in the stable reduction of $X_0(p^4)$, we find irreducible components, defined by $a^p-a=t^{p+1}$. These components are called Deligne-Lusztig curve for ${\rm SL}_2(\mathbb{F}_p).$ We also compute the intersection multiplicity datum in the stable reduction of $X_0(p^4)$.